THE EQUILIBRIUM EQUATIONS OF MEMBRANE SHELLS EXPRESSED IN GENERAL SURFACE COORDINATES

(1)

THE EQUILIBRIUM EQUATIONS OF MEMBRANE SHELLS EXPRESSED IN GENERAL SURFACE COORDINATES

G. DOl\WKOS Department of Mechanics, Technical University, H-1521, Budapest

Received June 2, 1989 Presented by Prof. Dr. J. Peredy

Ahstract

The aim of the paper is to derive the equations expressing the static equilibrium of membrane shells without introducing the bending theory. The first part gives a comprehensive introduction to the notions of tensor analysis which are needed in the forthcoming mechanical applications and contains a brief sketch of the background in classical differential geometry.

The derived formulas are illustrated on some simple examples in the last chapter.

Introduction

The stresses in membrane shells are usually determined by applying Pucher's differential equation, which enables us to calculate the projections of the stress components onto an extel'nal coordinate system. In this paper a general equation is pl'esented, by means of which the stress components can be expl'essed in an arbitl'ary surface coordinate system. This equilibrium equation is usually introduced as a special case of the bending theory. The aim of this paper is to derive the equation dil'ectly, applying as simple tools as possible. The geometry of the curved, two-dimensional sm'faces, as the mathematical background of the equilibrium equation, is discussed. The paper is in- tended to be a comprehensive introduction for graduate students in civil en- gineering and architecture.

Pucher's differential equation excellently demonstrates the fact, that the application of coordinate systems is of advantage when describing physical phenomena mathematically. This fact is generally accepted, but we must not forget, that the physical phenomena are totally independent of our coordinate systems. The representation of physical phenomena in coordinate systems may be regarded therefore as a disturbing type of description.

It is self-evident, that if a natm'allaw holds, its representation holds in an arbitrary coordinate system, as well. There are a couple of rules, which permit us to transform the representation of a natural law from a coordinate system into another one. If the coordinates of a phenomenon observed in several coordinate systems are transforming under these rules, the phenomenon is said to he coordinate-invariant or simply invariant.

(2)

86 G. DOJlOKOS

To eliminate the mentioned disturbing effect of the coordinate systems, basically two ways are possible:

The "direct" description, which doesn't use any coordinate systems. The technical application of this type of equations may be sometimes cum- bersome.

The formulation of general equations, where the form of the coordinate system itself is "blank". During technical applications any type of coordinate system may be "substituted" into the equation.

In the section 1.1 we 'will introduce the basic notions by the "direct"

way for the sake of comparison. Further on the second way will be followed and the equivalence ofthe two different descriptiom: "\Vill be indicated at appropriate places.

Acknowledgements

This paper was written by the author during his graduate studies at the Faculty of Architecture. The author expresses his gratitude to Prof. J. Peredy, who encouraged him re- peatedly to this work. Section 1.5 is a fruit of consultations with G. 3foussong. Conversations with J. Bali resulted the more exact mathematical formulation. The author expresses his thanks to Pr of J. Szenthe and }L Armuth for the careflll revision of the manuscript and is very indebted to his mother for the valuable technical help in the preparation of the material.

1. IHathematical preliminaries 1.1. Invar£ant quantit£es - tensors

1.1.1. Scalars

Physical phenomena are quantitatively described by numbers. In the simplest case a single numerical data identifies the observed variable. This type of physical variables are called scalars. For example: volume, mass, temperature.

If each point of the physical space is associated with a number, we arrive at a scalar field. For example if we measure the temperature at each point of a room, this data is represented by a scalar field. Two-dimensional scalar fields can be visualized as (generally curved) surfaces in the three-dimensional space.

Scalars can be interpreted as homogeneous, linear scalar-scalar functions.

A function is called homogeneous and linear if the following equations hold:

f(a) f(b) = f(a

+

^b) ⁽¹⁾

f(l.a) = If(a) (2)

(3)

EQUATIONS OF MEJIBRANE SHELLS 87

The scalar S defines for example the homogeneous linear scalar-scalar function h (x) = sx. For the sake of generality, scalars interpreted as homogeneous linear functions will be called Oth order tensors. The meaning of this ,viII (hopefully) become evident in the following sections.

1.1.2. Vectors

Physical quantities identified by a number and a direction are called vectors. For example: velocity, acceleration, force. Vectors may be visualized as directed intervals. The n-dimensional vector space is the set of all n-dimensional vectors, where vectors can be added ,vith each other in the usual way and can be multiplied by scalars. The three-dimensional Euclidean space is for example the vector space of the above-mentioned directed intervals. Two vectors are called equivalent in the Yector space, if they haye the same direction and magnitude.

If each point of the physical space is associated with a vector, we arrive at a vector field (not to be confused with the vector space). For example if we measme the magnitude and direction of velocity of the partieles on the sm- face of a streaming liquid, this set of data is represented by a t"wo-climensional vector field. As an other example we can measure the principal stresses at each point of a three-dimensional elastic continuum to arrive at three different three- dimensional vector fields.

Vectors can be interpreted as homogeneous linear scalar-vector or vector- scalar functions. Vector v defines for example the scalar-vector function (= vector-valued function with independent scalar variahle) g( x) = vx or the scalar-vector function scalar-valued function ,vith independent vector variahle) k(x) = vx. Vectors interpreted as homogeneous linear functions will he called 1st order tensors.

1.1.3. Second order tensors

Certain physical quantities can he descrihed neither hy scalars nor by vectors. This is the point, where second order tensors are introdUCed. Second order tensors are homogeneous linear vector-vector functions (= vector-valued functions with independent vector variahle). If each point of the physical space is associated with a second order tensor, we arrive at a tensor field. A 2nd order tensor field expresses the homogeneous linear connection hetween two vector fields. Second order tensors can hardly be directly visualised. We can form some image, no·wever, hy observing, that the application of a tensor to the unit sphere (formed by unit vectors of the three-dimensional Euclidean space) distortes the sphere into a general ellipsoid. During this transformation the unit vectors are rotated and their length changes, as well.

(4)

88 G. DOMOKOS

Two basic numbers are associated 'with a tensor: the order and the die mension. The order of the tensor fixes the number of vectors between which the tensor defines a functional relation. The dimension of the tensor fixes the dimension of the space where the above-mentioned vectors are interpreted.

To mention some examples for second order tensors:

The rotation tensor describes the rotation of a rigid body by defining a functional relation hetween the vectors associated with the points of the origie nal and the rotated body.

The planar state of stress is described by the stress tensor defining a functional relation between a direction vector and the stress vector in that direce tion.

1.2. Representation in coordinate systems

The n-dimensional base is a syfltem of n linearly independent vectors.

We ·will use mainly 2-and 3-dimensional bases. If a tensor is given \vith respect to a base, we speak about representation in a coordinate system.

1.2.1. Orthogonal systems

In the simplest case the vectors of our base ar~ mutually perpendicular unit vectors, this is called an orthogonal coordinate system. The base consisting of the vectors eel)' e(2)' and e(3) \"ill be denoted by K. The orthogonality of this base can be expressed by using the scalar (dot) product:

and

e(l)e(2)

=

^e(2)e(3)

=

^e(l)e(3)

=

0 The above equations can be expressed more concisely by

{

l i f i = k e(i)eO,) = 0 if i " k

(3)

(4)

(5) (In the forthcoming formulas the latin indices i, j, k, etc. are assumed to be equal to 1, 2 or 3.)

The symbol

(6) is commonly used and called the Kronecker-delta. According to (5) and (6) the Kronecker-delta is defined by

{

l i f i = k

(j. =

lk 0 if i " k (7)

(5)

EQUATIOSS OF ;yfKlfBRANE SHELLS 89

1.2.1.1. Vectors

Since the vectors of the base K are linearly independent, an arbitrary vector can be expressed with respect to this base as

(8) The set of numbers ^vi

=

(v_{l ,}_{V 2,V 3)} is called the coordinates of the vector v in the system K. We alTive at the geometrical interpretation of the coordinates if eq. (8) is multiplied by the vectors em' ("Multiplication" "\VilI mean, that we form the dot product ;v-ith each memher in the equation)

(9) The above formula contains three equations, depending on, which base vector was eq. (8) multiplied ,v-jth. Equation (9) demonstrates, that the coordinates l'i

are the orthogonal proj ections of the vector onto the coordinate axes.

In calculations the vector v is often substituted hy the coordinates Vi'

which doesn't mean, that tbe two things are identical. If we change the coordinate system, the coordinates change, but the vector doesn't. We introduce the notation

Vi

=

K(v) (10)

expressing, that Vi is the image of v in the system K.

1.2.1.2. Second order tensors

A second order tensor is uniquely given if we know the transformed version of an arbitrary vector v. It seems to be logical to deal \dth the transfor- mations of the base vectors first, since if

(11) and

(12) holds, than obviously

Tu = v (13)

where

(14) Multiplying (14) hy e(I()' we arrive at

(15)

(6)

90 G. DOMOKOS

The left hand side can be expressed by using (9) ~

(16) Introducing the notation

(17) we arrive at

(18) Forming a table with the values tUe in the following way is called the matrix of the tensor T in the system K, more concisely K(T):

(

t11 t12 t^13\

tif{ = t21 t22 t~3) t31 t32 t33

(19)

The determinant of the above matrix is called the determinant of the tensor T in the system K and is denoted by

det tile = t (20)

We ·will. calculate as an illustrative example the elements of the matrix of the planar rotation tensor F in the K system. The vector transformation can be written as:

a' = Fa (21)

With coordinates:

2

a~ = :2ftr.{Ja_p ⁽²²⁾

,,=1

(In the forthcoming formulas the greek indices

wm

be equal to I or 2.) Simi- larly to (17):

(23) Figure 1 demonstrates, that by rotating the base vectors of the planar system K we arrive at the vectors

I I '

e(l) = e(l) cos rp T e(2) srn rp

(24) e(2)

=

e(l) ( -sin rp)

+

^{e(2) cos}rp

According to this the representation of the tensor F in the system K is given by

(

COS rp sin rp )

h.

^p

=

K(F)

=

-sin rp cos rp Using (25) an arbitrary vector can be transformed in K.

(25)

(7)

EQU 4.TIONS OF ME?ifBRANE SHELLS 91

cosP

I / sinf

Fig. 1. The tensor of planar rotation

1.2.2. Sh:ew systems 1.2.2.1. Vectors

The vector v can be expressed not only in the system K, but in an arbitrary system A determined by the base vectors aUl on the condition, that the base vectors are linearly independent:

o

⁽²⁶⁾

V denotes the volume of the parallelepyds spanned by the three vectors. The

\'ector v can be expressed as

v =

::2

v iaU) (27)

l

Similarly to (10) Vi = A(v). In the skew system A the scalar product of the vector v and the base vectors isn't equal to the vector coordinates, since the scalar product of two different base vectors isn't zero. We ,vilI introduce therefore the reciprocal base aU) by

(28) ::Ylultiplying eq. (27) by 3,(I{) yields

(29) which indicates, that the coordinates of v in the system A are equal to the orthogonal projections to the reciprocal base in the proper scale. This is illustrated in the plane by Fig. 2.

Since the base vectors of the reciprocal system are linearly independent, v can be expressed as

(30) Multiplying (30) by aU) yields:

(31)

(8)

92 G. DOMOK06

Fig. 2. The skew reciprocal base

That means, that in the skew system A the vector v can be equally given by the numbers Vi or Vi' The numbers Vi ;v-ill. be called the contravariant coordi- nates, the numbers Vi the covariant coordinates of the vector v in the system A.

In the orthogonal system K the contTavariant and covaTiant coordinates naturally coincide. For further use we intToduce the notation

(32)

Let's now examine the geometTical inteTpretation of the symbols Vi and Vi, For the sake of simplicity we will work in the plane. Si milady to the notation introduced ahove, the Teciprocal vectors u(,,) "\\-ill. be denoted by a(O:). Rewriting now the two previous equations "'With the ne"w notations yields

(33) and

(34) Equation (34) contains the projections of v in the directions of the vectors a(o:) expressed in proper units. Since the vectOTs a(,,) aTen't unit vectors, we have to choose the quantity

I

^a(",)

I

^{as unit.}^Ifwe don't, than the magnitude of the projection may be calculated by

(35) Summarizing the above investigations: the covariant coordinates mean the orthogonal projections, the contravariant components the projections in the direction of the coordinate axes. We are now interested in the problem, how to

(9)

EQUATIONS OF ,1IEMBRANE SHELLS 93 calculate the covariant coordinates from the contravariant ones. Expressing v in both ways:

Let's multiply eq. (36) by a(i) yielding

vi{ =

.::f

via(i)a(i{) i

(36)

(37) The above equation demonstrates, that the connection between the two repre-

sentations is given by the scalar products of the base vectors. This products depend on t"WO indices, "WP. introduce the notation

(38) The numhers gu: are the elements of a matrix. Substituting (38) into (37) yields

(39) If the inverse of the matrix gil: exists (let's denote it hy

ik),

than it is easily derived, that

(40) There are some useful applications of the above derived results. Let's calculate the scalar product of two vectors in the skew coordinate system! We ,vill treat the follmving two Yectors:

The scalar product in contravariant representation:

" " i k U V

= ..:;;;;;,.

u v a(Oa(k)

ik

By using the formerly introduced gik notation:

" " i k

uv =..:;;;;;,. uv gik ik

Deriving the same expression by using the co variant components:

u v =

:2

^uiv"i^k

ik

Now let's substitute eq. (39) into eq. (43):

11 v

=

_."'=i

:5.'

uiv" _{ i

(41)

(42)

(43) (44)

(45)

(10)

94 G. DOMOKOS

and similarly:

(46) Equations (45) and (46) closely resemble to their analogons in orthogonal coordinate systems.

1.2.2.2. Second order tensors

We will proceed as we did in orthogonal systems. A second order tensor is given in the most natural and simple \I!ay if we know the h(i) transformed versions of the a(i) hase vectors. Knowing this vectors the transformation of an arbitrary vector may be executed, since if

and

then on the hasis of (13) and (14) ohviously Tu= v

" v = ~ ' " z,1·· < u(O

;

Let's multiply (50) 'with 1l(J;):

The left hand side can be written hecause of (31) and (32) as:

Let

According to this:

( 47)

(48)

(49)

(50)

(51)

(52)

(53)

(54) To calculate the quantities til; we used the hase vectors aU)' therefore the matrix tile "will he called the covariant representation of the tensor T in the skew system A. The contravariant representation t^ikmay be derived in a similar way. Remark, that transforming the components of the vector u by the matrices tik or t^ikwe always arrive at y components of different representa-

(11)

EQUATIONS OF 2HEMBR.4.iliE SHELLS 95 tion. To avoid this inconvenience let's introduce the mixed representation of the tensor T by

and

where naturally

and

i _ (i)

t./{ - a hUe)

v· _!

(55)

(56)

(57)

(58) (59) (60) The sequence order of the indices is not indifferent, since the matrices

t:

_i^!

and

t/

are in general not identical. \,Ve will introduce now the so-called Einstein summation convention for the dummy indices:

(61) If the dummies are the indices of a mixed representation tensor, then this summation is called the contraction of the tensor resulting a tensor of order 0, that means, a scalar. This operation is equivalent to the summation of the components in the main diagonal.

In generality: by summing a tensor of order n to a single pair of dummy indices we arrive at a tensor of order (n-2).

Let's examine now the meaning of the symbols ^gi!:and ^gikintroduced in eq. (38). By comparing (53) with (38) we can observe that the symbol g!!c

is the covariant representation of the E unit tensor, since on the basis of the equations (3) and (38) this tensor maps the base vectors onto themselves.

Similarly the symbol

i

^kis the contravariant representation of the unit tensor.

In the orthogonal system we havc naturally

Ik ~

gik == g

==

^Ui/{ (62)

The symbols glk and gi/{ are usually called the components of the metric tensor.

This name will be explained later.

We are going to investigate the relationship between the four possible representations of a second order tensor T in the skew system A. Since the various representations can be computed by using the covariant and contravariant base vectors, the transformation rules between the representations can

(12)

96 G. DOJJOKOS

be derived from the relation between the base vectors. Let's express the vectors a(i) as a linear combination of the vectors a(i):

(i) ik

a

=

ⁿ ^a(k) ⁽⁶³⁾

By multiplying eq. (63) with a^U) we arrive at

ilc ili

n =g (64)

According to this the correspondence between the two systems is given by (a)

(b)

(65)

Equation (65) enables us to determine the relation hetween the different tensor representations. ^TLet: ^.

tik = a(i)h(k)

If we substitute (65jh) into (66), we arrive at

tik = gn,a (i) h(f{)

On the hasis of (55):

(66)

(67)

(68) The relation hetween two arhitrary representations of the tensor T may be derived in a similar way. Remark, that the multiplication 'with

i"

^or^gik^results

the "moving" of an index up or dovv"11, respectively. Applying this to the metric tensor G:

(69) Developing this equation for two dimensions we arrive at the follo"\ving formulas:

1, 21 g12

g-=g = - - (70)

g where g denotes the determinant according to (20).

1.2.3. Curvilinear systems

The location of a point in space may be identified not only by the coordinates introduced before, but by the means of other parameters, as well. We ,till use the parameters 0i • Let A(x)

=

^Xithe representation of the position vector x in the skew system A. The functional relation

Xi

=f(0)

(71)

(13)

EQUATIONS OF MEMBRANE SHELLS 97 has to exist. Let's consider the parameters 6; as the coordinates of the position vector x. If the function

f

is non-linear, then the quantities 6_iare called the curvilinear coordinates of x. This fact will be denoted by

(72) The transformation of the coordinates to the system A^Gcan be carried out only in the case if the function

f

is invertible, i.e. the mapping is unique both ways. The curvilinear coordinate systems are often applied, for example the spherical coordinate system, called the spatial polar system, as well. For this special case eq. (71) may be 'written as

Xl

=

r cos cp sin y (73)

X 2 = r sin (p sin y

X3 = r cos y

In the system A^Gcorrespond to the constant value of any single parameter a curved surface in the three-dimensional space. If two parameters are simulta- neously constant, then we arrive at space curves (lines) after which the A^G system was named. ("Curvilinear system") The system A^Gcan be treated locally as a skew system. In other words, the system A^Gdefines a skew system A at each point of the three-dimensional space. The base vectors of the local system A are given by the tangent vectors of the coordinate lines

(H) (The derivation of vector fields will be discussed in section 4.2 in detail.) Up to now we were dealing with the curvilinear representation of the position vector x, hut this doesn't answer the question ahout the curvilinear representation of an arbitrary vector v \,ith origin differing from the origin of the coordinate lines. For convenience we define the curvilinear representation AG(v) as the representation of v in the skew system A determined hy the system A^Gat the origin of v hy the equation (74). Remark, that the representation of the metric tensor depends on the coordinates, as well

ox ox

go: = 06; 06" ⁽⁷⁵⁾

1.3. Transformation of coordinates

Our aim is to determine the transformation rules for tensor coordinates if we s"\Vitch from system A^Gto

A:G.

The hase vectors are those defined hy eq. (74):

7

(14)

G. DOMOKOS

OX

aU)= (J0

1

and the base vectors of the system

A

^G

where

x =/(0i )

0i = h(e_i)

Let's express the vectors ii(i) as the linear combinations of the vectors a(i):

ii(i) = p{a(j)

(76)

(77)

(78) (79)

(80) According to this equation the matrix f3 (which is quadratic, of course) inherits the first index from the original system, the second one (with-) from the transformed system. Let

On the basis of (80) and (81):

i -i-

V = V a(i) = v a(i)

Ca) Vi

= vi'pf,

(b) Vi =v_kf3'rc

(81)

(82)

\Ve can determine the relation between the representations gik and

gi!;

of the metric tensor:

(83)

and on the basis of (80):

(84) Based on the above formulas the general transformation rule for the coordinates of a second order tensor t is easily derived. Let

and on the basis of (80) and (82):

a)

tile = ^a(i)h(l;) tile

=

ii(i)h(!;)

(85)

(86)

(15)

EQUATIO;VS OF JfEJIBRA,,\'E SHELLS 99 Remark, that eq. (86) serves in many cases as an alternative definition for tensors. In practical applications we can decide often on the basis of this formula, whether the examined phenomenon can be described by a tensor or not.

We have to measure in an experiment the coordinates in two different coordinate systems, and if the measured quantities transform under the rules prescrib- ed by eq. (86), then they are the representations of a tensor. The connection between the two coordinate systems is given by the matrix

fJ.

Based on the equations (76), (77), (78) and (79) we can define now the components of the matrix

fJ

by the equations

and

Of course,

holds, as well.

fJr

=

ae"

ae

i

(87)

(88)

(89) The above given definitions may be easily generalized for contravariant and mixed representations, the determinant of "'which is an invariant scalar. It is worth observing, that in the equations (80) and (82) the relation between the ,-ector coordinates is the opposite of that between the vectors. This is the origin of the name "contravariant".

As an illustrative example for the transformation rules we

""ill

^calculate

the relation between the tensor coordinates given in an orthogonal system by eq. (17) and in a skew system by eq. (53) in two dimensions. Coordinates are illustrated in Fig. 3.

:~'C?S"

e11)

Fig. 3. Relative position of the planar K and A system In the examined planar case

(90) holds. Let

(91) 7*

(16)

100 G. DOJfOKOS

The elements of the transformation matrix {3 are readily derived as:

{3f = em = cos rp (3f = sin g'

a(l)

(3~ = 1 Now we can calculate the components of the matrix

tu,:

= cos (P <12 T sill (P ^t21

= cos (P t21

+

^sin^(p^t12

R2(R1t I R2. ) _ Vi fJi 21 T fJi&22 - sin rp cos cp( t12

>4('0') _ : _ R1(R1 I {<2 ) I {<2(R1

J:1 - 22 - [22 - IJ'i fJZhl T IJ'it12 T fJ'i fJZhl

(92)

(93)

By this example we wanted to underline, that the computation of the tram- formation has to do only \vith the different representations of the same physical quantity in different reference frames. If a tensor equation holds, than it holds in an arbitrary coordinate system, but in each system the form of the equation will be different, according to the rules of transformation derived in this section.

1.4. Differentiation of tensor fields 1.4.1. Scalar field

1.4.1.1. Directional derivative

For the sake of simplicity we will treat a two-dimensional scalar field, which can be \isualized as a curved surface in the three-dimensional space.

This surface will be denoted by Z.

We ,,,ill investigate the surface at point

P,

which corresponds to the point P of the scalar field. If we intersect the surface Z by a plane passing through

P

and orthogonal to S (the plane, on which we interprete the scalar field), then the result is a curve on the surface. The intersection of this 'orthogonal plane vvith S is a straight line, which

,,,ill

be denoted bye. Let us proceed now on e by the distance c. The value of the scalar field will be denoted by z', at the

(17)

EQUA:rIOiVS OF 2HEJJBRANE SHELLS 101

Fig. 4. Scalar field as curved surface

original point P ·with z. Now the directional derivative of the scalar field at point P in the direction e is defined by

This can be visualized as the directional tangent of the surface curve at point j5 in the plane of intersection. This is illustrated in Fig. 5:

Remark, that the directional derivative ha;:: been introduced "without the use of any coordinate systems.

/ '/ / /

/

Fig. 5. Directional derivative of scalar field

(18)

102 G. DO.HOKOS

1.4.1.2. Partial derivative

We will now use in the plane a coordinate system xC,,), In this system the scalar field can be interpreted as a scalar function in two variables in the form z = f(x(",»). If we calculate the directional derivatives in the directions of the coordinate lines, we arrive at the expressions

--=Z'et 8z

aXC",)

(95) which ,,,ill be called the partial derivatives. The quantities Z'''' can be represented by two Ecalar fields. (The partial derivatives of an n-dimensional scalar field are represented by n i'eparate scalar fields in n dimeni'ions.)

1.4.1.3. Gradient field

Despite the fact, that the partial derivatives of a scalar field depend un

the coordinate system, we are able to define a coordinate-invariant quantity ,dth the aid of them. Let's regard the partial derivatives as the components of a vector given in the same coordinate system as the original scalar field. \Ve can decide, whether they are actually vector coordinates by the transforma- tions rules derived in eq. (82jh). In the original system we have

z,_ = = - -Clz

~ ClXe,,) (96)

Transforming no",- to the new coordinate system

x

^bythe eq. (88) we arrive at z,;: = r

~z

⁼ ^r^8z

~~(P)

^=- ::;"J3g

ox"") oXC/l) OX(2) -

(97)

This illustrates, that the partial derivatives transform under the rule for vector coordinates. The physical invariant vector determined by the partial derivatives ,,,ill he called the gradient of the scalar field. The gradient of an n-dimensional scalar field is an n-dimensional vector field. The gradient vector field

",ill be denoted by g.

We ,dli try to visualize the gradient field in two dimensions. Figure 6 demonstrates a two-dimensional scalar field as a curved surface z

=

^f(^x,^y).

At point P of the surface the tangents parallel to the coordinate planes are indicated.

This tangents determine the tangent plane P 1 P ~P 3' The partial derivatives are the directional tangents of the lines e1 and e_2,therefore

(98)

(19)

EQUATIONS OF JfE.lIBRA5E SHELLS 103

Fig. 6. The gradient

The tangent of the interval PIP? on the plane xv is OP¹• If we measure the

~ - -' OP?

vector g from the orthogonal P' projection of the point

P,

then we find, that g is orthogonal to PIP 2' since the tangent of g can be expressed as

OP3 OP,;!

OP₃ OP_I

and is found to be the reciprocal value of the tangent of P₁P_2•The vector g indicates at each point the direction and magnitude of the fastest rate of change of the scalar field.

1.4.2. Vector field

1.4.2.1. Directional derivative

We will consider a three-dimensional vector field. This field defines the vector vat point P. Now we select an arbitrary straight line (direction) e pass- sing through P and proceed along this line by a distance e to arrive at point Pi. The vector defined by the vector field at this last mentioned point villI be denoted Vi. The directional derivative of the vector field at point P in the direction e is then defined by

lim (v' - v)

(99)

e-O e

(20)

104 G. Dml,WKOS

Fig. 7. Directional derivative of vector field This is illustrated by Fig. 7.

Remark, that the directional derivative of the vector field has been introduced

"without the use of any coordinate systems, similarly to the directional derivative of the scalar field. The directional derivative of an n-dimensional vector field is a vector field of the same dimension.

1.4.2.2. Partial derivati1:e

The vector field will be interpreted in the cOOTdinate system Xi' In this system the vector field can be interpreted as a vector-vector function in one variable, "iuce the vector v is the function of the position vector x, both vectors given "with their coordinates. We are going to determine the directional derivatives in the directions of the coordinate lines. In order to do this, we can write the vector field in the form

(100) on the basis of equations (27) ~1lld (32). Differentiating eg. (100) by th;> jth variahle "we arrive at

V'j = (viaU))'j Applying the rule for product differentiation yields

a)

or resoh-ed to covariant components b) V'j = vl'jaU

) - vIaU\

(101)

(102)

The first member contains the partial derivatives of the scalars Vi and VI

multiplied by the base vectors. This partial derivatives can be determined on the basis of section 1.4.1.

The second member contains the partial derivatives of the base vectors multiplied by the scalars v' and ^Vi'In a straight (orthogonal or skew) coordinate system this derivatives disappear, of course, since the base vectors are of constant magnitude and mrection. To "visualize the meaning of the second member, let's regard Fig. 8:

(21)

EQUA.TIONS OF 11,fE2\-fBRANE SHELLS 105

b) c)

Fig. 8. Connection between vectors and vector coordinates

It can be observed, that in a straight coordinate system the change of the vector coordinates sufficiently describes the change of the vector, therefore the first member of the partial derivative contains enough information. In curvilinear systems this is not the case:

The vectors in Fig. 8/b are equal, but their components aren't. In Fig.

Sic the opposite happens, the vectors are not equal, but their components are. In section 1.4.1.2. v,-e didn't meet this "second member", hecause the representation of scalar fields is independent of the base vectors.

1.4.2.3. ChristojJel symbols

In section 1.4.2.1. we observed, that the directional derivative of a vector field is a vector field, as well. According to this the vector field aU)' j can be resolved to components in the hases a(i) or a(i):

(103) Multiplying the above equation by a(h) we arrive at

r

^(I)

r

^-I

r

aCi)'jaCk)

=

îj1a âU;)⁼ îj1fh⁼ îjk ^(10-1)

Equations (103) and (104) are equivalent definitions for the quantities

r

^with

three indices. The quantities

r

_ijk^and

r;j

are called the Christoffel symbols of the first and second kind, respectively. They 'were introduced by the mathematician Elvin Bruno Christoffel. The

r

symbols are cube-matrices, with 27 components in 3 dimensions. The first index of the Christoffel symbol refers to the variable (base vector field) to be differentiated, the second tells, in which direction it has to be differentiated and the third identifies the component of the directional derivative resolved already in the coordinate system. We are no\y going to derive some useful formulas in connection \\>ith Christoffel symbols.

Multiplying (103) 'ivith the base vectors we arrive at

(105)

(22)

106 G. DOJfOKOS

This illustrates, that the third index of the Christoffel symbols can he

"moved" up or down by the method first described in eq. (68). This does not hold for the first two indices, since the Christoffel symbols are not third order tensors.

Differentiating eq. (74) by the jth variable yields

(106) which displays the symmetry of the Christoffel symbols '\\ith respect to the first two indices. Differentiating eq. (38) by the kth variable yields

(107) Comparing this with (103) we find, that

(108) Writing the above result cyclically thrice, we arrive at the equation system

(109) b)

r

kij

r

ijl{ = gjk'i

c)

r

^jki

r

^ijk⁼ ^gki'j

Composing now the equation (b/)

+

^{(cl) -} ^(a/)^yields

(lIO) This formula is convenient when calculating the Christoffel symbols in a coordinate system where the coordinates of the metric tensor are already knO'Vll.

By using eq. (70) we arrive at

d)

r

^{t _}^{i r - - -}¹ ^og - - r - - = - _ _ ^og ^_ ¹

^olr-

^gT

2g

8g

is

8e Vg oe

^(lIl)

by using the result

b) og .

=gglS

ogis

1.4.2.4. Tensor derivative

Applying the symbols introduced in (104) to (102) yields

(lI2) By changing the indices of the last two members we arrive at

(lI3)

(23)

EQUATIONS OF MEMBRANE SHELLS 107

Writing the equation for the components only (by multiplying with a(k») (114)

Vdj

⁼ ^{Vi'j -}

v/ir;j

The above expression will be called the covariant derivative of the vector field v. The covariant derivatives Vifj are analogons to the partial derivatives of the scalar field. They depend on the choice of coordinate system, but, as we did with the partial derivatives, it is possihle to define a coordinate-inya- riant quantity by using them as coordinates. This invariant \vill he called the second order tensor deriYative of the vector field. The second index of the tensor derivative is always covariant, this is the reason to call the quantities v'i j

the covariant derh-atives. This characteristic was inherited from the fact, that we interpreted the position vector x in a contravariant (traditional) way.

Resolving the position vector into covariant coordinates we could derive the contravariant derivative, hut this has no practical reason.

In the case of the gradient vector we proved, that the partial derivatives of the scalar field transform under the rule prescrihed for vector coordinates.

Now we are going to do the same for the coordinates of the second order tensor derivative. In order to do this we calculate the covariant derivative of the vector y in the system a(i~:

Applying the "chain rule" yields

Comparing the right hand side of the ahove two equation yields v-!"a(i) = 1'.1 .a(i)fJ~

1 I} 'I J 1

Multiplying this with ^a(t;)=

fJ¥.a u{)

we arrive at

'-1- - .

I

fJ"fJ

^j

Vk,j - vie j It .7

(lIS)

(116)

(117)

(lIS) Comparing the ahove equation , ... ith (86th) we see, that our statement is proven, the covariant derivatives are actually the coordinates of a tensor.

Summarizing our results we can say, that the covariant differentation is an analogon of the partial differentiation and differs from the latter only in the curvilinear representation of tensor fields of order higher than zero.

1.4.3. Second order tensor field

We are going to introduce the derivative of the second order tensors. On the hasis of the previously deriYed equations it ,vill not he quite surprising, that the derivative of a second order tensor field is a third order tensor field.

(24)

108 G. DOJYWKOS

We are going to deliver a rather formal description, but later we -will examine the derivatives of specific second order tensor fields.

}Iultiplying the covariant tir representation of the tensor T with the contra- variant vector components u' and v' we arrive at the scalar s:

(119) Differentiating the above expression by the kth variable and considering eq. (1l4) yields

(120)

~ i I j . ' .J i ir! j i l r i

~;J'u ,/.v ,.. t;,u V !/. - t,.,ll /;1 V - t;J'u, V 1-:1

• I i. • J I • J •

This can be written in the folIo'wing form, as ",-eH:

(121) by accepting the foIlo'wing definition

(122) Changing the dummy indices we arrive at the form

(123) This equation holds for an arbitrary matrix tij and given components u' and v^Jif and only if

(124) The above expression is the definition of the covariant derivative of the tensor T in the representation t_{ij •}This can be expressed hy

(125) As mentioned before, we assume, that the quantItIeS

tulle

represent a third order tensor. To prove this, we use the same procedure as we did in the equations (115)-(118):

T _'k_-=t-~I-·a_l].It^! ^{-(I) -(j)}'" _-

(H)

m _ _l',Ie=T

,!.--=tu,·a

, ii<k) a

I

" ⁽ⁱ⁾^a

(j)fJH

.. ^f, t-71~ _lJ_!..iiU)iiU) =

t,1

.a(i)aU)IB~

'J ll, !..

t;]

I

^{k =} tij

I kfJffJJfJ~

(126) (127) (128) (129)

(25)

EQUA.TIONS OF j"fEZ,IBRANE SHELLS

Similarly to eq. (124) we arrive at the following expressions:

i

I

ⁱ ^I ^{I r i} ^{i r l}

t'j "

=

^t.j ,,, T t'_j Icl - t'l jk

t

JI

t·j tJr^l I t.lr^j i le = i 'k - I i/( T i k(

.i] I - t^ij ^-Ltljrⁱ -L tilr'i

~ k - 'k I lel I lel

109

(130) (131) (132) Based on our experiences with second order tensors we can generalize for higher order ones: The covaI"iant derivative of an nth order tensor in a given representation can be computed by calculating the partial derivatives of the scalar tensor components and adding n members with Christoffel symbols.

The result is an (n 1) th order tensor.

1.4.4. The Riemann-Christoffel tensor

Equation (U8) demonstrates, that the components of the covariant derivative of the vector v are the representation of a second order tensor. In eq.

(124) the covariant derivative of second order tensors is introduced. Based on this, we are going to execute covariant differentiation on the second order tensor deriyative of v. Let

(133) Applying eq. (124) to the above expression yields

(134) We want to investigate, whether the indices in the covariant derivative can be changed or not, in other words, whether Vdjk =

vd

^kjor not. In the case simple partial differentiation this can be done. The quantities

vii

_{, J}k' can be expressed by the simple change of indices:

Expressing now the difference of the investigated quantities we arrive at

vil

jk - V;jkj = V;'jk - V;'kj - Vm'kr~

+

(135)

(136)

Since the indices in the simple partial derivatives can be changed, the first two term8 cancel each other and finally we have

I ^! ^(rm ^rm ^rmrl ^rrnr!)

Vi jk - Vi!kj = Vrn ik'j - ij'k - lj iT; - lle ;j (137) The left hand side of the above equation is obviously the representation of a third order tensor. This fact implies, that the bracet on the right hand side has to be the representation of a fourth order tensor, the first index of which is contravariant and the follo"wing three covariant.

(26)

110 G. DOMOKOS

Up to now we had to do only with tensors of order equal or lower than three. The appearance of a fourth order tensor doesn't imply difficulties, because all our former definitions for tensors are easily generalized. Returning now to eq. (137), let's denote the fourth order tensor by

(138) The quantItIes r}~k will be called the representation of the fourth order Riemann-Christoffel tensor. Now it is easy to answer our pre"dous question:

the indices of the covariant derivative can be changed if and only if

(139) The Riemann- Christoffel ten::,or is of course invariant under the transformation of coordinates, therefore if an equation holds in an arbitrary coordinate system, then it holds in each one. If we choose the orthogonal coordinate system, then eq. (139) is trivial, therefore it holds always. Now "WC have to ask, whether the orthogonal coordinate system exists in the examined space or not.

This is not a trivial question, since in an equivalent way we may ask, ,"'hether the structure of the examined space satisfies the euclidean axioms, or not.

The two dimensional case is discussed in the following section. The three di- mensional case goes beyond the range of this paper, but remark, that the first man to estabilish a non-euclidean geometry ,vithout contradictions was J{mos Bolyai. His geometry is the so-called hyperbolic geometry. Later the elliptic geometry was elaborated. In the hyperbolic geometry the curvature of space is a negath-e constant, in the elliptic geometry a positive constant.

The most general geometry is due to Bernhard Riemann. In the Riemann geometry the curvature of space is non-constant. The general relativity theory of Albert Einstein was based on the Riemann geometry.

This illustrates, that the Riemann-Christoffel tensor is closely related to the curyature of space, therefore it is called the Riemann-Christoffel curvature tensor.

1.5. Geometry of wrved surfaces

,'\Then Carl Friedrich Gauss was asked to partICIpate in the geodesic surveying of the county Hannover, the great germ an mathematician medita- ted for a long time over the sufficient and necessary condition of the existence of a measure-preserving planar map of a hilly landscape. His investigations resulted in one of the most outstanding theorem of his career, he himself called it "Theorema egregium". He proved, that at each point of the surface a scalar quantity can be calculated which is invariant under the transformation of coordinates. This scalar is now called the Gauss-curyature of the surface.

The necessary and sufficient condition for the surface to havc a measure-pre-

(27)

EQUATIOSS OF JIEMBRA.YE SHELLS 111 serving map in the euclidean plane is the disappearance of the Gauss curvature. It is surprising, that despite the fact, that the surface is enbedded in the three-dimensional euclidean space, it is theoretically possible, to measure the Gauss curvature "in the surface" for example flat, two-dimensional creatures moving exclusively in the surface could do that.

If the Gauss curvature doesn't disappear, then the surface can't be map- ped in a measure-preserving way onto the euclidean plane, that means, that the euclidean geometry doesn't hold on the surface. In this section ·we will try to get acquainted with the intristic geometry of this non-euclidean surfaces.

1.5.1. Interpretation of the metric tensor

We will investigate the geometrical meaning of the metric tensor introduced in eq. (38). Vie consider a plane with coordinates x(~) and the infinitesimal line element ds will he resolved to components in this coordinate system.

ds = dxCf.a(~) (140\

We vvill now multiply ds with itself arriving at

(141) which is the square of the length of the line element. Equation (141) is a straight- forward generalization of the Pythagoras formula, in differential geometry it is called the first fundamental form of the surface. In the usual orthogonal coordinate system the coordinates of the metric tf'nsor are l'Pprp,,('ntNI hy tIll' unit matrix, and the "well-known form of the Pythagoras theorem holds. If we introduce an other coordinate 8ystem in the plane and transform the components of the metric tensor under the rule given in eq. (86) then the validity of the euclidean geometry will not be disturbed.

Orthogonal coordinate 8ystems are, all the same, equivalent to any other coordinate system, therefore we can prescribe, in which arbitrary coordinate system we wish the metric tensor to be represented hy the unit matrix.

1.5.2. Classification of two-dimensional surfaces

Now we ask the inyerse question as before: what happenes, if we define in a region of the plane the components of the metric tensor arbitrarily in a gi"'en coordinate system, and there is no coordinate transformation, under which the representation becomes identic with the unit matrix? In this case the given g metric tensor field defines a non-euclidean geometry in the plane.

This can be visually realized by bending the plane into the three-dimensional plane. This bending must include stretching, as well. This is the reason, why

(28)

112 G. DO.'fOKOS

non-euclidean surfaces are often called curved surfaces. This name refers to the enbedding of a surface v"ith non-euclidean metric into a higher dimensional euclidean space. If the mentioned bending doesn't include stretching, then we arrive at the well-known developable surfaces with euclidean metric.

We conclude from this, that the metric uniquely determines the geometry of the surface, but it doesn't uniquely determine the form of the surface in the embedding euclidean space.

It is hard to visualize curved spaces if their dimension is higher than two, because for the visualization we need the embedding euclidean space, the dimension of which is always higher than the dimension of the curved space. In the case described just before the embedding space had one dimension more than the curved surface. This is not bound to be so, since a one-dimensional 'wire can be bent in a 'way, that it can't be embedded in a two-dimensional surface. (Remark, that the intristic geometry of a wire doesn't change by bending.)

The two-dimensional surfaces v,ill be classified on the bases of the mi- nimally necessary dimension of the embedding euclidean space. If this dimension is two, then the surface is called a plane, if it is three, then the surface is called a hypersurface, if it is larger than three, then it is called a general two- dimensional surface. In this general case the intristic geometry of the surface is described by the Riemann-Christoffel tensor. To calculate the components of this tensor we need the coordinates of the metric tensor and their derivatives only, therefore we can say, that the intristic geometry of the surface is comple- tely described by the metric tensor. However, this calculations are rather cum- bersome, so it is difficult to see the connection between the given metric tensor field and the intristic geometry.

We are now especially interested in the description of hypersurfaces, which is a special case. The intristic geometry of a hy-persurface can be described by a tensor field, which is much simpler than the Riemann-Christoffel tensor, but can't be applied to general two-dimensional surfaces. We are going to get acquainted with this simpler tensor field.

1.5.3. The second order curvature tensor

We are going to investigate a region of a two-dimensional hypersurface embedded in the three-dimensional euclidean space, with coordinate system

xC,,) in the surface. At point P we can regard the tangent plane of the surface

and the normal vector of the tangent plane. This normal vector is called the normal vector of the surface at point P. With the aid of this we are able to define a normal vector field a(3) with

(142)

(29)

EQUATIOiYS OF JIEJIBR..J.YE SHELLS 113

The orthogonality condition with the hase vectors yields

(143) The direction of the a(3) vector field depends on the sUl'face coordinates, but not the magnitude. Therefore the partial derivatives of the unit normal vectol' field are surface Yector fields, that can hc resolved in the surface coordinate system:

(144) Multiplying this with aCYl ,,-e arrive at

(145) The quantities b"p are the representation of a second order tensor, this can be demonstrated hy the transformation equations. Nmr \',-e will derive some useful formulas in connection with this tensor. Differentiating (143) and hy using (145) we arriYe at

ah)'pa(3) = - a(,,)a(3)'p = bp" (146)

Writing eC[. (104.) in the ahove introduced coordinate system yields a

r (;,)

^I

r

^(.1)

(")'P

=

^"p'la ^T ^"P3^U ⁽¹⁴⁷⁾

On the hasis of the two previous equations we have

(148) Using the derived formulas for the Christoffel symhols the last equation can he re-formulated as

-r,,3p ⁽¹⁴⁹⁾

The tensor B represented hy the matrix b"p will he called the second order curvatlll'e tensor. Other representations of the tensor are found hy multiplication with the metric tensor:

(150) The mixed representation can he derived hy interpreting eq. (144) in an other representation:

(151) From the ahove equation follows, that

d d ^{_x -} l (f-)d ^.C!-

a(3) = a(3)'" x - - }"pa x (152)

8

(30)

114 C. D01IOKOS

Let's multiply the infinitesimal vector with the line element ds by using equations (140) and (152):

d _{a(3) S -}a - _- b _~fJa(lJ)a·" x a(y) a·;' --x - - b "flu;, ~fJd ~d x x ;- da(3)ds = - b"fJdx~dx;'

(153) (154) In classical differential geometry the right hand side of eq. (154) is called the second fundamental form of the surface. The coefficients 611' b₁₂= b_{21 ,}b₂₂ were denoted by E, F, G by Carl Friedrich Gauss.

/x-

/

A

8 ^~x:

Fig. 9. Connection between the curvature and the unit normal .-cctor

We will now try to visu alize the components of the second order curvature tensor in mixed representation. We assume, that all components but

bi

disappear, and "we intersect the surface with a normal plane along a ^X(l)line.

This is demonstrated in Fig. 9.

Since a(3) = 1, the length of the infinitesimal vector

d a(3) a(3)'1

a

x ^I - ^b¹la(l)

a·

x ¹ (155) is equal to the angle dq; between the surface normals at point A and B. If this angle is divided by the length of the vector a(1)dx^l, then we arrive at the curvature of the surface line ^X(l),which is equal to the curvature of the surface in the direction X(I).

dcp = i da(3) I as I all) ldxl

Now let's assume, that

bi =

0 hut

bi

^7'- ⁰

Figure 10 illustrates, that

da(3)

=

a(3)'1 dx¹

= -

bia(2)aX¹ so the twist of the surface is

d{}

ds

I

da(3)

I

^a(l)dx'

I

(156)

(157)

(158)

(31)

EQU.tTIOSS OF .IIE.UBRASE SHELLS 115

Fig. 10. Conllcction hetwecn thc twist and thc unit normal vector

If a(l) = ^a(2)'then

bi

⁼ ^b~and both components are equal to the t"wist of the surface along the coordinate lines. If the opposite holds, that the two compo- nf'nts aren't equal, hut they are closely related by eq. (158) to the twist.

1.5.4. Covariant derivative of sll1face tensor fields

A general vector field with origin on the surface can he resoh·ed into in- surface and normal components:

v = via(i) = v"a(::<)

+

^v₃^a(3) ⁽¹⁵⁹⁾

This equation is differentiated according to eq. (113), yielding

V ^! a(3) 3 i 13

We conclude from equations (143) and (144), that

a(3h,a(3) = 0 therefore on the basis of (105)

Similarly

(160)

(161)

(162)

(163) The previous equations demonstrate, that the Christoffel symbols disappear on the surface, if they have more than one index equal to three. Using this fact and equations (148) and (160) ·we arrive at

(164) and

(165) 8*